Districting and Gerrymandering Andrea Scozzari University Niccol` o Cusano Caen, July 8-12 2014 Andrea Scozzari University Niccol` o Cusano Districting and Gerrymandering Caen, July 8-12 2014 1 / 69
Definitions Political Districting (PD) consists of subdividing a given territory into a fixed number of districts in which the election is performed. A given number of seats, generally established on the basis of the population of the district, is allocated to each district. These seats must be assigned to parties within the district according to the adopted electoral system that rules out how the citizens’ votes are transformed into seats. The PD problem has been studied since the 60’s and many different models and techniques have been proposed with the aim of preventing districts’ manipulation to favor some specific political party ( gerrymandering ). Andrea Scozzari University Niccol` o Cusano Districting and Gerrymandering Caen, July 8-12 2014 2 / 69
Definitions Given the vote distribution, different district plans may reverse the outcome of an election (see R.J. Dixon and E. Plischke (1950), American Government: Basic Documents and Materials, New York, Van Nostrand.) Neutral district plans are necessary to oppose partisan manipulation of electoral district boundaries ( gerrymandering ) The aim is to provide automatic procedures for political districting, designed so as to be as neutral as possible Andrea Scozzari University Niccol` o Cusano Districting and Gerrymandering Caen, July 8-12 2014 3 / 69
Example 1. A territory divided into 81 elementary units (sites) of equal population 2. Each site is colored yellow (40 sites) or orange (41 sites) that constitute the vote distribution 3. 9 (uninominal - 1 seat at stake) districts must be drawn, each formed by 9 contiguous sites (perfect population equality) Andrea Scozzari University Niccol` o Cusano Districting and Gerrymandering Caen, July 8-12 2014 4 / 69
Example Try to make Yellow/Orange party win as many seats as possible!! by drawing 9 districts Andrea Scozzari University Niccol` o Cusano Districting and Gerrymandering Caen, July 8-12 2014 5 / 69
Example Try to make Yellow/Orange party win as many seats as possible!! by drawing 9 districts Yellow party wins 8 seats, Orange party wins 1 seat! Andrea Scozzari University Niccol` o Cusano Districting and Gerrymandering Caen, July 8-12 2014 5 / 69
Example Orange party wins 8 seats, Yellow party wins 1 seat! Andrea Scozzari University Niccol` o Cusano Districting and Gerrymandering Caen, July 8-12 2014 6 / 69
Gerrymandering This was what happened in Massachusetts in 1821 when the governor Elbridge Gerry drew the electoral districts in order to be re-elected. In this way, he was able to take advantage from the territorial subdivision in order to gain seats. This bad malpractice is known as gerrymandering from a particular salamander-shape of one of the districts obtained. Andrea Scozzari University Niccol` o Cusano Districting and Gerrymandering Caen, July 8-12 2014 7 / 69
Presidential election USA 2004: Pennsylvania Districts 12 Andrea Scozzari University Niccol` o Cusano Districting and Gerrymandering Caen, July 8-12 2014 8 / 69
Presidential election USA 2004: Pennsylvania Districts 12 Andrea Scozzari University Niccol` o Cusano Districting and Gerrymandering Caen, July 8-12 2014 9 / 69
Presidential election USA 2004: Pennsylvania Districts 12 Andrea Scozzari University Niccol` o Cusano Districting and Gerrymandering Caen, July 8-12 2014 10 / 69
Presidential election USA 2004: Pennsylvania Districts 12 Andrea Scozzari University Niccol` o Cusano Districting and Gerrymandering Caen, July 8-12 2014 11 / 69
Presidential election USA 2004: Pennsylvania Districts 12 Andrea Scozzari University Niccol` o Cusano Districting and Gerrymandering Caen, July 8-12 2014 12 / 69
Political Districting Criteria 1. Integrity : Each territorial unit belongs to only one district and it cannot be split between two different districts. 2. Contiguity : A district is formed by a set of geographically contiguous units. Andrea Scozzari University Niccol` o Cusano Districting and Gerrymandering Caen, July 8-12 2014 13 / 69
Political Districting Criteria 3. Absence of enclaves : No district can be fully surrounded by another district. 4. Compactness : The districts must have regular geometric shapes. Octopus- or banana-shaped districts must be avoided. Andrea Scozzari University Niccol` o Cusano Districting and Gerrymandering Caen, July 8-12 2014 14 / 69
Political Districting Criteria 5. Population equality : Districts populations must be as balanced as possible. Andrea Scozzari University Niccol` o Cusano Districting and Gerrymandering Caen, July 8-12 2014 15 / 69
Political Districting Criteria 5. Population equality : Districts populations must be as balanced as possible. There are other PD criteria that are seldom used, among the others we mention: - the respect of natural boundaries - fair representation of ethnic minorities - respect of integrity of communities . . . Andrea Scozzari University Niccol` o Cusano Districting and Gerrymandering Caen, July 8-12 2014 15 / 69
Political Districting Indicators There is the need to define correct indicators to measure the above criteria. Compactness The compactness of a district depends on its area, on the distances between territorial units and the district center, the perimeter, the geometrical shape, its length and its width, the district population, and so on. 1. Dispersion measures: district area compared with the area of canonical compact figure (for example the circle); 2. Perimeter based measures: perimeter compared with area; 3. Population measures: district population compared with the population of the smallest compact figure (for example a circle) which contains the whole district. Andrea Scozzari University Niccol` o Cusano Districting and Gerrymandering Caen, July 8-12 2014 16 / 69
Political Districting Indicators This is a measure of compactness of a district D obtained by computing the percentage of sites in the circle centered in the center s of radius d is that do not belong to D (Arcese, Battista, Biasi, Lucertini, Simeone, 1992). Andrea Scozzari University Niccol` o Cusano Districting and Gerrymandering Caen, July 8-12 2014 17 / 69
Political Districting Indicators The above index can be refined so as to evaluate compactness also with respect to population since each territorial unit can be weighted by its population. Let P d h be the total population of the units within the circle of radius d , then the compactness index is: K P d h − P h � P d h h =1 K = the total number of districts P h = the population of district h Andrea Scozzari University Niccol` o Cusano Districting and Gerrymandering Caen, July 8-12 2014 18 / 69
Political Districting Indicators Moment of Inertia Let c be a point in district D . The moment of inertia of district D with respect to c is the weighted sum of the squared distances of all territorial units in D from c . The weight of each distance is given by the population of the corresponding territorial unit. The moment of inertia is minimized by setting c equal to the center of gravity g of the district. n h � p i · ( d g h i ) 2 i =1 n h = number of units in district h d g h = the distance between unit i in h and the center of gravity of h i p i = population in the territorial unit i Andrea Scozzari University Niccol` o Cusano Districting and Gerrymandering Caen, July 8-12 2014 19 / 69
Political Districting Indicators Population Equality The most popular indexes of population equality are global measures of the distance between the populations of the districts and the mean district population P . K � | P h − P | h =1 K K = the total number of districts P h = the population of district h Andrea Scozzari University Niccol` o Cusano Districting and Gerrymandering Caen, July 8-12 2014 20 / 69
Political Districting Indicators Population Equality Other indexes can be built simply by replacing the L 1 norm by other norms: K ( P h − P ) 2 � h =1 K K = the total number of districts P h = the population of district h Andrea Scozzari University Niccol` o Cusano Districting and Gerrymandering Caen, July 8-12 2014 21 / 69
Political Districting Indicators Population Equality Unfortunately, the range of these measures depends on the size of the total population, so relative measures with values in the [0 , 1] interval are usually preferred (Arcese, Battista, Biasi, Lucertini, Simeone, 1992): K � | P h − P | h =1 2( K − 1) P K = the total number of districts P h = the population of district h Andrea Scozzari University Niccol` o Cusano Districting and Gerrymandering Caen, July 8-12 2014 22 / 69
Political Districting Indicators Population Equality A very different index is given by the inverse coefficient of variation (ICV) (Shubert and Press, 1964) � K � ( P h P − 1) 2 � � � h =1 � K K = the total number of districts P h = the population of district h Andrea Scozzari University Niccol` o Cusano Districting and Gerrymandering Caen, July 8-12 2014 23 / 69
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